Abstract
After briefly considering the ancient Greek and nineteenth-century history of incommensurables (magnitudes that do not have a common aliquot part) and incomparables (magnitudes such that the larger can never be surpassed by any finite number of additions of the smaller to itself), this paper undertakes two tasks. The first task is to consider whether the numerical accommodation of incommensurables by means of the extension of the ordered field of rational numbers to the field of reals is `similar' or analogous to the numerical accommodation of incomparables by means of the extension of the ordered field of reals to the field of hyperreals. The second task is to evaluate several contemporary attempts to use concepts and techniques of the nonstandard mathematics of hyperreals to address classical, Zenonian puzzles concerning continuous magnitudes. The result of both these undertakings is, in a certain sense, `deflationary'.
Citation
Michael White. "Incommensurables and Incomparables: On the Conceptual Status and the Philosophical Use of Hyperreal Numbers." Notre Dame J. Formal Logic 40 (3) 420 - 446, Summer 1999. https://doi.org/10.1305/ndjfl/1022615620
Information